5(x - 4) = 5x - 20 - Aurero

Understanding the Context

What Is 5(x - 4) = 5x - 20?

This equation demonstrates the application of the distributive property in algebra. On the left side, 5 is multiplied by each term inside the parentheses:
5(x - 4) = 5 × x – 5 × 4 = 5x – 20
So, expanding the left side gives us:
5x – 20 = 5x – 20

At first glance, the equation appears balanced for all values of x. But let’s take a closer look to confirm this logically and mathematically.

Key Insights

Why the Equation Holds True for All x

To verify, we can simplify both sides:

• Start with: 5(x - 4)
  
• Apply distributive property: 5x – 20
  
• Now the equation becomes: 5x – 20 = 5x – 20

Both sides are identical, meaning every real number for x satisfies this equation. This makes the equation an identity, not just a linear equation with one solution.

Why does this matter?

Final Thoughts

Understanding that 5(x - 4) = 5x – 20 is an identity helps reinforce the concept of equivalent expressions—critical for algebraic fluency. It also prevents common mistakes, such as assuming there is a unique solution when, in fact, infinitely many values of x make the equation true.

Solving the Equation Step-by-Step

Although this equation has no unique solution, here’s how you would formally solve it:

1. Expand the left side:
   5(x – 4) = 5x – 20
   
2. Substitute into the original equation:
   5x – 20 = 5x – 20
   
3. Subtract 5x from both sides:
   5x – 20 – 5x = 5x – 20 – 5x
   → –20 = –20
   
4. This simplifies to a true statement, confirming the equation holds for all real numbers.

If this equation had a solution, we would isolate x and find a specific value. But because both sides are identical, no single value of x satisfies a “solution” in the traditional sense—instead, the equation represents a consistent identity.